- #1
TheMathNoob
- 189
- 4
Homework Statement
Let A1,A 2,...be an infinite sequence of events such that A1⊂A2⊂... Prove that
Pr(∪Ai)=limn→infPr(An)
∪ is also an iterator that starts from i=1 to infinity. How can you put those iterators?[/B]
Homework Equations
I decided to use induction
The Attempt at a Solution
Proof by induction
Base case
let A1⊂A2[/B]
so A1∪A2=A2
Therefore Pr(A1∪A2)=Pr(A2)
Inductive Step
let Pr(∪Ai)=limk→infPr(Ak)
then we have to show that if Ak⊂Ak+1 then limk→infPr(Ak∪Ak+1)=limk→infPr(Ak+1)
so
This is true because Ak∪Ak+1=Ak+1
so limk→infPr(Ak∪Ak+1)=limk→infPr(Ak+1)
so by math induction Pr(∪Ai)=limn→infPr(An)
is that right?